ONE-WAY ANALYSIS OF VARIANCE (ANOVA)

 

Purpose

 

To test the affect that one categorical variable (IV or SV) has on another continuous variable (DV). A 1-way ANOVA is used to compare three or more population means, m, using sample means,.

 

BACKGROUND

 

1.                  Participants can be randomly chosen from the population or obtained by some other (more convenient) means (e.g., haphazard/convenient sampling).

2.                  Participants are either randomly assigned to 1 level of the independent variable, such as in an experimental design, or can be categorized based on membership to some type of subject variable (academic major, for example), such as in a non-manipulated/correlational design. This type of assignment would be a between-participants/independent groups design.

·         It is also possible that participants experience all levels of an independent variable (within-participants/repeated measures design).         

 

THE NULL HYPOTHESIS

 

The null hypothesis for ANOVA states that the population means from which the samples are taken are equal.  In other words:

 

m1= m2 = m3 =…= mk

 

The mean of the population, m, refers to the hypothetical value that would result if all the individuals from the population had been observed. The null hypothesis states that there is at least 1 pair of means that are different. It does not state that all means are different. To determine which means are different you should conduct a post-hoc analysis.

 

LOGIC OF THE TEST

 

An ANOVA tests mean differences among 3 or more groups by analyzing the variance in the data collected.  The estimate for the total amount of variance in the data can be divided into estimates for between and within groups variance.

 

Estimate of the total variance in the data

=

Estimate of the between subjects variance

+

Estimate of the within subjects variance

 

 

·         The estimate of the total variance in the data is the average distance from each score to the grand mean (X - m).

·         The estimate of the between subjects variance is the average distance from the group means to the grand mean (- m). It is the variance that we can explain -- it is good variance.

·         The estimate of within subjects variance is the average distance from each score to the group mean (X - ). It is the variance that we can't explain -- it is bad variance.

 

The F-value is the ratio of between groups variance to within groups variance.  Conceptually, the F ratio can be thought of using the following formula:

 

 

Ideally the F-ratio would be large, the proportion of good variance should be higher than the proportion of bad variance. Each of the estimates are calculated by dividing the Sum of Squares by the appropriate degrees of freedom (k -1 & N - k)

 

THE FORMULA

 

 

; ; dfB = k – 1

 

; ; dfW = N – k

 

Where  n is the number of participants per group

            N is the number of participants in the entire sample

            k is the number of groups

           

THE SUMMARY TABLE

 

As you complete the calculations it is helpful to complete a summary table. You should total the SS and df columns. You should place an asterisk (*) next to the F-ratio if it is statistically significant.

 

 

Source             SS                    df                     MS                   F          h2

 

Between

 

Within

_______________________________

 

Total

 

STATING A CONCLUSION

 

Your conclusion should contain several pieces of information. First, state if the alternative hypothesis was supported (if null was rejected) or refuted (if null was retained). Next state the statistics, both descriptive (i.e., means) and inferential (i.e., F-ratio, df, alpha level) using proper APA format. Third, tell if any causal relationship exists (only when null is rejected and experimental design used). Lastly, determine the effect sizes and discuss the strength of the relationship between the variables. Example:

 

The hypothesis that type of beverage consumed was related to perceived attractiveness was supported. The omnibus F test of perceived attractiveness ratings for participants who consumed non-alcoholic beer (M = 7), regular beer (M = 8) and water (M = 4) was significant, F (2, 9) = 6.49, p < .05. The type of beverage caused differences in perceived attractiveness because type of beverage was manipulated. The strength of the relationship between type of beverage and perceived attractiveness was medium (h2 = .59).